Estimation of Population Mean and Proportion Notes

Estimation of Population Mean and Proportion

  • Definition of Estimation

    • Estimation refers to the assignment of value to a population parameter based on the corresponding sample statistics.
    • It commonly involves calculating a margin of error to assess the reliability of estimates pertaining to the population.

  • Confidence Levels and Confidence Intervals

    • Confidence Level: Represents the percentage of time that an estimated interval will capture the true population parameter.
    • Confidence Interval: An interval constructed around the point estimate that signifies how much confidence we have that it contains the true population parameter.
    • It is usually expressed as a percentage, indicating the degree of certainty in our estimates.
    • Denoted mathematically as:
      ext{Confidence Level} = (1 - ext{alpha}) imes 100\%
      where alpha is the level of significance.

  • Mathematical Formulas for Estimation

    • Estimation of Population Mean:
    • Formula for required sample size to estimate the population mean with a given margin of error (E): n = rac{(z_{α/2})^2 imes ext{σ}^2}{E^2} where
      • z_{α/2} is the critical value (z-score) corresponding to the confidence level.
      • ext{σ} is the population standard deviation.
      • E is the margin of error.
    • Determining Critical Values:
    • Example values for common confidence levels:
      • 90% confidence level: z = 1.645
      • 95% confidence level: z = 1.96
      • 99% confidence level: z = 2.576

  • Margin of Error in Estimation

    • The margin of error is crucial in estimating both the mean and the proportion; it dictates how accurate our estimates are compared to the true population parameter.

  • Estimation of Population Proportion:

    • Formula for determining sample size when estimating population proportion: n = rac{(z_{α/2})^2}{4E^2} where
      • E is the margin of error for the population proportion estimation.

  • Example Problem: Determining Sample Size

    • Scenario: To ensure a 90% confidence level that the estimation error does not exceed 0.5 units with a standard deviation of 2.
    1. Standard Deviation: 2
    2. Margin of Error: 0.5
    3. Using the formula:
      n = rac{(1.645) imes 2}{0.5} leads to a result that needs to be squared to find the final sample size, approximately yielding 43.29, which rounds to 43.
    • Further Problem: Finding sample size for a 99% confidence to estimate a proportion of 0.2
    1. Using the formula:
      n = rac{(2.576^2)}{4 imes (0.02^2)} which provides the required sample size.

  • Laboratory Activity:

    • Requires students to complete activities using Excel to find sample sizes based on given parameters including confidence levels, margin of error, and standard deviation, and is submitted through JASP.